Abstract
In the present paper we show a dichotomy theorem for the complexity of polynomial evaluation. We associate to each graph H a polynomial that encodes all graphs of a fixed size homomorphic to H. We show that this family is computable by arithmetic circuits in constant depth if H has a loop or no edges and that it is hard otherwise (i.e., complete for VNP, the arithmetic class related to #P). We also demonstrate the hardness over ℚ of cut eliminator, a polynomial defined by Bürgisser which is known to be neither VP nor VNP-complete in \(\mathbb F_2\), if VP ≠ VNP (VP is the class of polynomials computable by arithmetic circuits of polynomial size).
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de Rugy-Altherre, N. (2012). A Dichotomy Theorem for Homomorphism Polynomials. In: Rovan, B., Sassone, V., Widmayer, P. (eds) Mathematical Foundations of Computer Science 2012. MFCS 2012. Lecture Notes in Computer Science, vol 7464. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32589-2_29
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DOI: https://doi.org/10.1007/978-3-642-32589-2_29
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